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Search: a037233 -id:a037233
Displaying 1-10 of 11 results found. page 1 2
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A062318 Numbers of the form 3^m - 1 or 2*3^m - 1; i.e., the union of sequences A048473 and A024023. +10
43
0, 1, 2, 5, 8, 17, 26, 53, 80, 161, 242, 485, 728, 1457, 2186, 4373, 6560, 13121, 19682, 39365, 59048, 118097, 177146, 354293, 531440, 1062881, 1594322, 3188645, 4782968, 9565937, 14348906, 28697813, 43046720, 86093441, 129140162 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
WARNING: The offset of this sequence has been changed from 0 to 1 without correcting the formulas and programs, many of them correspond to the original indexing a(0)=0, a(1)=1, ... - M. F. Hasler, Oct 06 2014
Numbers n such that no entry in n-th row of Pascal's triangle is divisible by 3, i.e., such that A062296(n) = 0.
The base 3 representation of these numbers is 222...222 or 122...222.
a(n+1) is the smallest number with ternary digit sum = n: A053735(a(n+1)) = n and A053735(m) <> n for m < a(n+1). - Reinhard Zumkeller, Sep 15 2006
A138002(a(n)) = 0. - Reinhard Zumkeller, Feb 26 2008
Also, number of terms in S(n), where S(n) is defined in A114482. - N. J. A. Sloane, Nov 13 2014
a(n+1) is also the Moore lower bound on the order of a (4,g)-cage. - Jason Kimberley, Oct 30 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, Properties of Hesse derivatives of cubic curves, arXiv:2309.05048 [math.AG], 2023. See p. 9.
Gyula Tasi and Fujio Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).
FORMULA
a(n) = 2*3^(n/2-1)-1 if n is even; a(n) = 3^(n/2-1/2)-1 if n is odd. - Emeric Deutsch, Feb 03 2005, offset updated.
From Paul Curtz, Feb 21 2008: (Start)
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3).
Partial sums of A108411. (End)
G.f.: x^2*(1+x)/((1-x)*(1-3*x^2)). - Colin Barker, Apr 02 2012
a(2n+1) = 3*a(2n-1) + 2; a(2n) = ( a(2n-1) + a(2n+1) )/2. See A060647 for case where a(1)= 1. - Richard R. Forberg, Nov 30 2013
a(n) = 2^((1+(-1)^n)/2) * 3^((2*n-3-(-1)^n)/4) - 1. - Luce ETIENNE, Aug 29 2014
a(n) = A052993(n-1) + A052993(n-2). - R. J. Mathar, Sep 10 2021
E.g.f.: (1 - 3*cosh(x) + 2*cosh(sqrt(3)*x) - 3*sinh(x) + sqrt(3)*sinh(sqrt(3)*x))/3. - Stefano Spezia, Apr 06 2022
a(n) = (1/3)*([n=0] - 3 + (1+(-1)^n)*3^(n/2) + ((1-(-1)^n)/2)*3^((n+1)/2)). - G. C. Greubel, Apr 17 2023
EXAMPLE
The first rows in Pascal's triangle with no multiples of 3 are:
row 0: 1;
row 1: 1, 1;
row 2: 1, 2, 1;
row 5: 1, 5, 10, 10, 5, 1;
row 8: 1, 8, 28, 56, 70, 56, 28, 8, 1;
MAPLE
A062318 :=proc(n)
if n mod 2 = 1 then
3^((n-1)/2)-1
else
2*3^(n/2-1)-1
fi
end proc:
seq(A062318(n), n=1..37); # Emeric Deutsch, Feb 03 2005, offset updated
MATHEMATICA
CoefficientList[Series[x^2*(1+x)/((1-x)*(1-3*x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 20 2012 *)
A062318[n_]:= (1/3)*(Boole[n==0] -3 +3^(n/2)*(2*Mod[n+1, 2] +Sqrt[3] *Mod[n, 2]));
Table[A062318[n], {n, 50}] (* G. C. Greubel, Apr 17 2023 *)
PROG
(Magma) I:=[0, 1, 2]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2) -3*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Apr 20 2012
(PARI) a(n)=3^(n\2)<<bittest(n, 0)-1 \\ [Program corresponds to offset=0, a(0)=0, a(1)=1.] - M. F. Hasler, Oct 06 2014
(SageMath)
def A062318(n): return (1/3)*(int(n==0) - 3 + 2*((n+1)%2)*3^(n/2) + (n%2)*3^((n+1)/2))
[A062318(n) for n in range(1, 41)] # G. C. Greubel, Apr 17 2023
CROSSREFS
Cf. A062296, A024023, A048473, A114482. Pairwise sums of A052993.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), this sequence (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A037233 (actual order of a (4,g)-cage).
Smallest number whose base b sum of digits is n: A000225 (b=2), this sequence (b=3), A180516 (b=4), A181287 (b=5), A181288 (b=6), A181303 (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10).
KEYWORD
nonn,easy
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001
EXTENSIONS
More terms from Emeric Deutsch, Feb 03 2005
Entry revised by N. J. A. Sloane, Jul 29 2011
STATUS
approved
A054760 Table T(n,k) = order of (n,k)-cage (smallest n-regular graph of girth k), n >= 2, k >= 3, read by antidiagonals. +10
22
3, 4, 4, 5, 6, 5, 6, 8, 10, 6, 7, 10, 19, 14, 7, 8, 12, 30, 26, 24, 8, 9, 14, 40, 42, 67, 30, 9, 10, 16, 50, 62 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,1
REFERENCES
P. R. Christopher, Degree monotonicity of cages, Graph Theory Notes of New York, 38 (2000), 29-32.
LINKS
Table of n, a(n) for n=0..31.
Andries E. Brouwer, Cages
M. Daven and C. A. Rodger, (k,g)-cages are 3-connected, Discr. Math., 199 (1999), 207-215.
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cubic Cages
Gordon Royle, Cages of higher valency
Pak Ken Wong, Cages-a survey, J. Graph Theory 6 (1982), no. 1, 1-22.
FORMULA
T(k,g) >= A198300(k,g) with equality if and only if: k = 2 and g >= 3; g = 3 and k >= 2; g = 4 and k >= 2; g = 5 and k = 2, 3, 7 or possibly 57; or g = 6, 8, or 12, and there exists a symmetric generalized g/2-gon of order k - 1. - Jason Kimberley, Jan 01 2013
EXAMPLE
First eight antidiagonals are:
3 4 5 6 7 8 9 10
4 6 10 14 24 30 58
5 8 19 26 67 80
6 10 30 42 ?
7 12 40 62
8 14 50
9 16
10
CROSSREFS
Moore lower bound: A198300.
Orders of cages: this sequence (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).
Graphs not required to be regular: A006787, A006856.
KEYWORD
nonn,tabl,nice,hard,more
AUTHOR
N. J. A. Sloane, Apr 26 2000
EXTENSIONS
Edited by Jason Kimberley, Apr 25 2010, Oct 26 2011, Dec 21 2012, Jan 01 2013
STATUS
approved
A000066 Smallest number of vertices in trivalent graph with girth (shortest cycle) = n.
(Formerly M1013 N0380) 		+10
16
4, 6, 10, 14, 24, 30, 58, 70, 112, 126 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
Also called the order of the (3,n) cage graph.
Recently (unpublished) McKay and Myrvold proved that the minimal graph on 112 vertices is unique. - May 20 2003
If there are n vertices and e edges, then 3n=2e, so n is always even.
Current lower bounds for a(13)..a(32) are 202, 258, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072. - from Table 3 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012
Current upper bounds for a(13)..a(32) are 272, 384, 620, 960, 2176, 2560, 4324, 5376, 16028, 16206, 49326, 49608, 108906, 109200, 285852, 415104, 1141484, 1143408, 3649794, 3650304. - from Table 3 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012
REFERENCES
A. T. Balaban, Trivalent graphs of girth nine and eleven and relationships among cages, Rev. Roum. Math. Pures et Appl. 18 (1973) 1033-1043.
Brendan McKay, personal communication.
H. Sachs, On regular graphs with given girth, pp. 91-97 of M. Fiedler, editor, Theory of Graphs and Its Applications: Proceedings of the Symposium, Smolenice, Czechoslovakia, 1963. Academic Press, NY, 1964.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Table of n, a(n) for n=3..12.
Gabriela Araujo-Pardo, Geoffrey Exoo, and Robert Jajcay, Small bi-regular graphs of even girth Discrete Mathematics 339.2 (2016): 658-667.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Brendan McKay, W. Myrvold and J. Nadon, Fast backtracking principles applied to find new cages, 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, 188-191.
M. O'Keefe and P. K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory, B 29 (1980), 91-105.
Gordon Royle, Cubic Cages
Eric Weisstein's World of Mathematics, Cage Graph
Pak Ken Wong, Cages-a survey, J. Graph Theory 6 (1982), no. 1, 1-22.
FORMULA
For all g > 2, a(g) >= A027383(g-1), with equality if and only if g = 3, 4, 5, 6, 8, or 12. - Jason Kimberley, Dec 21 2012 and Jan 01 2013
CROSSREFS
Cf. A006787, A052453 (number of such graphs).
Orders of cages: A054760 (n,k), this sequence (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).
KEYWORD
nonn,hard,more,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
Additional comments from Matthew Cook, May 15 2003
Balaban proved 112 as an upper bound for a(11). The proof that it is also a lower bound is in the paper by Brendan McKay, W. Myrvold and J. Nadon.
STATUS
approved
A184940 Irregular triangle C(n,g) counting the connected 4-regular simple graphs on n vertices with girth exactly g. +10
8
1, 1, 2, 5, 1, 16, 0, 57, 2, 263, 2, 1532, 12, 10747, 31, 87948, 220, 803885, 1606, 8020590, 16828, 86027734, 193900, 983417704, 2452818, 11913817317, 32670329, 1, 152352034707, 456028472, 2, 2050055948375, 6636066091, 8, 28466137588780, 100135577616, 131 (list; graph; refs; listen; history; text; internal format)
OFFSET
5,3
COMMENTS
The first column is for girth exactly 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4. The row length is incremented to g-2 when n reaches A037233(g).
LINKS
Table of n, a(n) for n=5..41.
Jason Kimberley, Incomplete table of i, n, g, C(n,g)=a(i) for row n = 5..36
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g
EXAMPLE
1;
1;
2;
5, 1;
16, 0;
57, 2;
263, 2;
1532, 12;
10747, 31;
87948, 220;
803885, 1606;
8020590, 16828;
86027734, 193900;
983417704, 2452818;
11913817317, 32670329, 1;
152352034707, 456028472, 2;
2050055948375, 6636066091, 8;
28466137588780, 100135577616, 131;
CROSSREFS
Connected 4-regular simple graphs with girth at least g: A184941 (triangle); chosen g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184943 (g=3), A184944 (g=4), A184945 (g=5), A184946 (g=6).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), this sequence (k=4), A184950 (k=5), A184960 (k=6), A184970 (k=7), A184980 (k=8).
KEYWORD
nonn,hard,tabf
AUTHOR
Jason Kimberley, Feb 24 2011
STATUS
approved
A184941 Irregular triangle C(n,g) counting the connected 4-regular simple graphs on n vertices with girth at least g. +10
7
1, 1, 2, 6, 1, 16, 0, 59, 2, 265, 2, 1544, 12, 10778, 31, 88168, 220, 805491, 1606, 8037418, 16828, 86221634, 193900, 985870522, 2452818, 11946487647, 32670330, 1, 152808063181, 456028474, 2, 2056692014474, 6636066099, 8, 28566273166527, 100135577747, 131 (list; graph; refs; listen; history; text; internal format)
OFFSET
5,3
COMMENTS
The first column is for girth at least 3. The row length sequence starts: 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4. The row length is incremented to g-2 when n reaches A037233(g).
LINKS
Table of n, a(n) for n=5..41.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
EXAMPLE
1;
1;
2;
6, 1;
16, 0;
59, 2;
265, 2;
1544, 12;
10778, 31;
88168, 220;
805491, 1606;
8037418, 16828;
86221634, 193900;
985870522, 2452818;
11946487647, 32670330, 1;
152808063181, 456028474, 2;
2056692014474, 6636066099, 8;
28566273166527, 100135577747, 131;
CROSSREFS
Connected 4-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184940 (triangle); chosen g: A184943 (g=3), A184944 (g=4), A184945 (g=5), A184946 (g=6).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: A185131 (k=3), this sequence (k=4), A184951 (k=5), A184961 (k=6), A184971 (k=7), A184981 (k=8).
KEYWORD
nonn,hard,tabf
AUTHOR
Jason Kimberley, Jan 10 2012
STATUS
approved
A191595 Order of smallest n-regular graph of girth 5. +10
7
5, 10, 19, 30, 40, 50 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
Current upper bounds for a(8)..a(20) are 80, 96, 124, 154, 203, 230, 288, 312, 336, 448, 480, 512, 576. - Corrected from "Lower" to "Upper" and updated, from Table 4 of the Dynamic cage survey, by Jason Kimberley, Dec 29 2012
Current lower bounds for a(8)..a(20) are 67, 86, 103, 124, 147, 174, 199, 230, 259, 294, 327, 364, 403. - from Table 4 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012
LINKS
Table of n, a(n) for n=2..7.
M. Abreu et al., A family of regular graphs of girth 5, Discrete Math., 308 (2008), 1810-1815.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
G. Royle, Cages of higher valency
FORMULA
a(n) >= A002522(n) with equality if and only if n = 2, 3, 7 or possibly 57. - Jason Kimberley, Nov 02 2011
CROSSREFS
Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), this sequence (n,5).
Moore lower bound on the orders of (k,g) cages: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306(k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10),A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 02 2011
KEYWORD
nonn,more,hard
AUTHOR
N. J. A. Sloane, Jun 07 2011
EXTENSIONS
a(2) = 5 prepended by Jason Kimberley, Jan 02 2013
STATUS
approved
A218553 Order of (5,n) cage, i.e., minimal order of 5-regular graph of girth n. +10
7
6, 10, 30, 42 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
a(7) <= 152, a(8) = 170, a(12) = 2730. - From Royle's page via Jason Kimberley, Dec 21 2012
LINKS
Table of n, a(n) for n=3..6.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency
Eric Weisstein's World of Mathematics, Cage Graph (claims too much)
FORMULA
a(n) >= A061547(n+1).
CROSSREFS
Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), this sequence (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).
KEYWORD
hard,more,nonn
AUTHOR
Arkadiusz Wesolowski, Nov 02 2012
EXTENSIONS
a(7) deleted by Jason Kimberley, Dec 21 2012
STATUS
approved
A218554 Order of (6,n) cage, i.e., minimal order of 6-regular graph of girth n. +10
7
7, 12, 40, 62 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
a(7) <= 294, a(8) = 312, a(12) = 7812. - From Royle's page via Jason Kimberley, Dec 26 2012
LINKS
Table of n, a(n) for n=3..6.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency
Eric Weisstein's World of Mathematics, Cage Graph (claims too much)
FORMULA
a(n) >= A198306(n).
CROSSREFS
Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), this sequence (6,n), A218555 (7,n), A191595 (n,5).
KEYWORD
hard,more,nonn
AUTHOR
Arkadiusz Wesolowski, Nov 02 2012
EXTENSIONS
a(7) deleted by Jason Kimberley, Dec 21 2012
STATUS
approved
A218555 Order of (7,n) cage, i.e., minimal order of 7-regular graph of girth n. +10
7
8, 14, 50, 90 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
a(8) <= 658, a(12) <= 32928. - Jason Kimberley, Dec 29 2012
LINKS
Table of n, a(n) for n=3..6.
Andries E. Brouwer, Cages
G. Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency
Eric Weisstein's World of Mathematics, Cage Graph (claims too much)
FORMULA
a(n) >= A198307(n).
CROSSREFS
Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), this sequence (7,n), A191595 (n,5).
KEYWORD
hard,more,nonn
AUTHOR
Arkadiusz Wesolowski, Nov 02 2012
EXTENSIONS
Edited by Jason Kimberley, Dec 21 2012
STATUS
approved
A185140 Irregular triangle E(n,g) counting not necessarily connected 4-regular simple graphs on n vertices with girth exactly g. +10
2
1, 1, 2, 5, 1, 16, 0, 58, 2, 264, 2, 1535, 12, 10755, 31, 87973, 220, 803973, 1606, 8020967, 16829, 86029760, 193900, 983431053, 2452820, 11913921910, 32670331, 1, 152352965278, 456028487, 2, 2050065073002, 6636066126, 8, 28466234288520, 100135577863, 131, 8020967, 16829 (list; graph; refs; listen; history; text; internal format)
OFFSET
5,3
COMMENTS
The first column is for girth at least 3. The column for girth g commences when n reaches A037233(g).
LINKS
Table of n, a(n) for n=5..43.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g
FORMULA
The n-th row is the sequence of differences of the n-th row of A185340:
E(n,g) = A185340(n,g) - A185340(n,g+1), once we have appended 0 to each row of A185340.
Hence the sum of the n-th row is A185340(n,3) = A033301(n).
EXAMPLE
05: 1;
06: 1;
07: 2;
08: 5, 1;
09: 16, 0;
10: 58, 2;
11: 264, 2;
12: 1535, 12;
13: 10755, 31;
14: 87973, 220;
15: 803973, 1606;
16: 8020967, 16829;
17: 86029760, 193900;
18: 983431053, 2452820;
19: 11913921910, 32670331, 1;
20: 152352965278, 456028487, 2;
21: 2050065073002, 6636066126, 8;
22: 28466234288520, 100135577863, 131;
CROSSREFS
Initial columns of this triangle: A185143 (g=3), A185144 (g=4).
KEYWORD
nonn,hard,tabf
AUTHOR
Jason Kimberley, Jan 06 2013
STATUS
approved
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